An improved bound for the Lyapunov rank of a proper cone

Given a proper cone K in Rn with its dual K∗ , the complementarity set of K is C (K) := {(x, s) : x ∈ K, s ∈ K∗ , x, s = 0}. A matrix A on Rn is said to be Lyapunov-like on K if Ax, s = 0 for all (x, s) ∈ C (K). The set of all such matrices forms a vector space whose dimension β (K) is called the Lyapunov rank of K. This number is useful in conic optimization and complementarity theory, as it relates to the number of linearly-independent bilinear relations needed to express the complementarity set. This article is a continuation of the study initiated in [6] and further pursued in [3]. By answering several questions posed in [3], we show that β (K) is bounded above by (n − 1) 2 , thereby improving the previously known bound of n2 −n. We also show that when β (K) ≥ n, the complementarity set C (K) can be expressed in terms of n linearly-independent Lyapunov-like matrices. Keywords Lyapunov rank · Perfect cone

• Type: Journal
• Year: 2016
• Where: OL
• Authors: Michael Orlitzky, M. Seetharama Gowda
• Keywords: OL 2016